The number of roots of following equation is:. sin(πx) = | log |x| |

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{need} \mathrm{to} \mathrm{draw} \mathrm{graph} \mathrm{of} \sin \left(\mathrm{\pi x}\right) \mathrm{and} \left|\log \left|\mathrm{x}\right|\right| \mathrm{to} \mathrm{find} \mathrm{number} \mathrm{of} \mathrm{solutions}. \\ \mathrm{Period} \mathrm{of} \mathrm{sinx} \mathrm{is} 2\mathrm{\pi }, \mathrm{hence} \\ \mathrm{Period} \mathrm{of} \sin \left(\mathrm{\pi x}\right) \mathrm{is} \frac{2\mathrm{\pi }}{\mathrm{\pi }}=2 \\ \mathrm{To} \mathrm{draw} \mathrm{graph} \mathrm{of} \left|\log \left|\mathrm{x}\right|\right|, \mathrm{follow} \mathrm{these} \mathrm{steps}. \\ 1. \mathrm{Draw} \mathrm{graph} \mathrm{of} \mathrm{logx} \\ 2. \mathrm{To} \mathrm{get} \log \left|\mathrm{x}\right|, \mathrm{take} \mathrm{mirror} \mathrm{image} \mathrm{of} \mathrm{graph} \mathrm{on} \mathrm{right} \mathrm{hand} \mathrm{side} \mathrm{of} \mathrm{y}-\mathrm{axis} \\ \mathrm{about} \mathrm{y} -\mathrm{axis}. \\ 3. \mathrm{To} \mathrm{get} , \left|\log \left|\mathrm{x}\right|\right| \mathrm{remove} \mathrm{take} \mathrm{mirror} \mathrm{image} \mathrm{of} \mathrm{graph} \mathrm{below} \mathrm{x}-\mathrm{axis} \mathrm{about} \\ \mathrm{x}-\mathrm{axis}, \mathrm{and} \mathrm{then} \mathrm{remove} \mathrm{graph} \mathrm{below} \mathrm{x}-\mathrm{axis}. \end{gathered}$$

$\mathrm{As} \mathrm{two} \mathrm{graphs} \mathrm{intersect} \mathrm{at} 20 \mathrm{points}, \mathrm{hence} \mathrm{there} \mathrm{are} 20 \mathrm{solutions}.$


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